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Macaulay2Doc :: nullhomotopy

nullhomotopy -- make a null homotopy

Description

nullhomotopy f -- produce a nullhomotopy for a map f of chain complexes.

Whether f is null homotopic is not checked.

Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.

i1 : A = ZZ/101[x,y];
i2 : M = cokernel random(A^3, A^{-2,-2})

o2 = cokernel | -45x2+43xy+25y2 27x2+45xy+7y2   |
              | -28x2-49xy+26y2 -21x2-16xy-19y2 |
              | -9x2-32xy-12y2  6x2+30xy+15y2   |

                            3
o2 : A-module, quotient of A
i3 : R = cokernel matrix {{x^3,y^4}}

o3 = cokernel | x3 y4 |

                            1
o3 : A-module, quotient of A
i4 : N = prune (M**R)

o4 = cokernel | -50x2+27xy-32y2 -19x2+xy+7y2 x3 x2y-8xy2+29y3 -48xy2+44y3 y4 0  0  |
              | x2-48xy-39y2    -36xy-5y2    0  -17xy2+46y3   49xy2-28y3  0  y4 0  |
              | 21xy-18y2       x2+28xy-10y2 0  -7y3          xy2-36y3    0  0  y4 |

                            3
o4 : A-module, quotient of A
i5 : C = resolution N

      3      8      5
o5 = A  <-- A  <-- A  <-- 0
                           
     0      1      2      3

o5 : ChainComplex
i6 : d = C.dd

          3                                                                              8
o6 = 0 : A  <-------------------------------------------------------------------------- A  : 1
               | -50x2+27xy-32y2 -19x2+xy+7y2 x3 x2y-8xy2+29y3 -48xy2+44y3 y4 0  0  |
               | x2-48xy-39y2    -36xy-5y2    0  -17xy2+46y3   49xy2-28y3  0  y4 0  |
               | 21xy-18y2       x2+28xy-10y2 0  -7y3          xy2-36y3    0  0  y4 |

          8                                                                         5
     1 : A  <--------------------------------------------------------------------- A  : 2
               {2} | -42xy2+16y3    20xy2-29y3 42y3      -12y3      -22y3      |
               {2} | -7xy2-2y3      -42y3      7y3       35y3       35y3       |
               {3} | 12xy+23y2      -19xy+41y2 -12y2     -41y2      19y2       |
               {3} | -12x2+4xy-34y2 19x2-50y2  12xy-27y2 41xy+26y2  -19xy-45y2 |
               {3} | 7x2+36xy+16y2  -43xy-9y2  -7xy-34y2 -35xy+16y2 -35xy+7y2  |
               {4} | 0              0          x+39y     34y        48y        |
               {4} | 0              0          44y       x-31y      -33y       |
               {4} | 0              0          19y       -17y       x-8y       |

          5
     2 : A  <----- 0 : 3
               0

o6 : ChainComplexMap
i7 : s = nullhomotopy (x^3 * id_C)

          8                             3
o7 = 1 : A  <------------------------- A  : 0
               {2} | 0 x+48y 36y   |
               {2} | 0 -21y  x-28y |
               {3} | 1 50    19    |
               {3} | 0 -46   -46   |
               {3} | 0 2     38    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                             8
     2 : A  <------------------------------------------------------------------------- A  : 1
               {5} | 43 9  0 -46y    29x-26y xy-8y2       5xy+11y2     5xy-12y2    |
               {5} | 3  40 0 16x-45y 13x+20y 17y2         xy+8y2       -49xy+14y2  |
               {5} | 0  0  0 0       0       x2-39xy-10y2 -34xy-39y2   -48xy-38y2  |
               {5} | 0  0  0 0       0       -44xy+28y2   x2+31xy-12y2 33xy-35y2   |
               {5} | 0  0  0 0       0       -19xy+43y2   17xy-4y2     x2+8xy+22y2 |

                   5
     3 : 0 <----- A  : 2
              0

o7 : ChainComplexMap
i8 : s*d + d*s

          3                    3
o8 = 0 : A  <---------------- A  : 0
               | x3 0  0  |
               | 0  x3 0  |
               | 0  0  x3 |

          8                                       8
     1 : A  <----------------------------------- A  : 1
               {2} | x3 0  0  0  0  0  0  0  |
               {2} | 0  x3 0  0  0  0  0  0  |
               {3} | 0  0  x3 0  0  0  0  0  |
               {3} | 0  0  0  x3 0  0  0  0  |
               {3} | 0  0  0  0  x3 0  0  0  |
               {4} | 0  0  0  0  0  x3 0  0  |
               {4} | 0  0  0  0  0  0  x3 0  |
               {4} | 0  0  0  0  0  0  0  x3 |

          5                              5
     2 : A  <-------------------------- A  : 2
               {5} | x3 0  0  0  0  |
               {5} | 0  x3 0  0  0  |
               {5} | 0  0  x3 0  0  |
               {5} | 0  0  0  x3 0  |
               {5} | 0  0  0  0  x3 |

     3 : 0 <----- 0 : 3
              0

o8 : ChainComplexMap
i9 : s^2

          5         3
o9 = 2 : A  <----- A  : 0
               0

                   8
     3 : 0 <----- A  : 1
              0

o9 : ChainComplexMap

Ways to use nullhomotopy :